Close Interstellar CometsPreprint typeset using LATEX style emulateapj v. 12/16/11

REALISTIC DETECTABILITY OF CLOSE INTERSTELLAR COMETS

Nathaniel V. Cook1, Darin Ragozzine2,3,4, Mikael Granvik5,6, Denise C. Stephens1

Close Interstellar Comets

ABSTRACT

During the planet formation process, billions of comets are created and ejected into interstellar space.The detection and characterization of such interstellar comets (also known as extra-solar planetesimalsor extra-solar comets) would give us in situ information about the efficiency and properties of planetformation throughout the galaxy. However, no interstellar comets have ever been detected, despitethe fact that their hyperbolic orbits would make them readily identifiable as unrelated to the solarsystem. Moro-Martın et al. 2009 have made a detailed and reasonable estimate of the properties of theinterstellar comet population. We extend their estimates of detectability with a numerical model thatallows us to consider “close” interstellar comets, e.g., those that come within the orbit of Jupiter. Weinclude several constraints on a “detectable” object that allow for realistic estimates of the frequencyof detections expected from the Large Synoptic Survey Telescope (LSST) and other surveys. Theinfluence of several of the assumed model parameters on the frequency of detections is explored indetail. Based on the expectation from Moro-Martın et al. 2009, we expect that LSST will detect0.001-10 interstellar comets during its nominal 10-year lifetime, with most of the uncertainty from theunknown number density of small (nuclei of ∼0.1-1 km) interstellar comets. Both asteroid and cometcases are considered, where the latter includes various empirical prescriptions of brightening. Usingsimulated LSST-like astrometric data, we study the problem of orbit determination for these bodies,finding that LSST could identify their orbits as hyperbolic and determine an ephemeris sufficientlyaccurate for follow-up in about 4–7 days. We give the hyperbolic orbital parameters of the mostdetectable interstellar comets. Taking the results into consideration, we give recommendations tofuture searches for interstellar comets.Subject headings: circumstellar matter – comets: general – Kuiper Belt – minor planets, asteroids –

planetary systems – solar system: formation

1. INTRODUCTION

The current understanding of planet formation sug-gests that very large numbers of minor bodies are ejectedinto interstellar space by planets during and after for-mation (e.g., Safronov 1972; Duncan et al. 1987; Doneset al. 2004). In typical simulations of solar system forma-tion, only a small fraction of the small bodies that havea close encounter with the giant planets are capturedinto the Oort cloud (the current source of long-periodcomets, Oort 1950), the rest are ejected into the inter-stellar medium. Despite changing understanding of theformation and properties of our Oort cloud (e.g., Lev-ison et al. 2010a; Kaib et al. 2011), extra-solar debrisdisks (e.g., de Vries et al. 2012), and planet formation(e.g., Chiang & Laughlin 2012), there is general con-sensus that interstellar space must be populated witha non-trivial population of small bodies, including thosecorresponding in size to the asteroids and comets in thesolar system.

nathanielvcook@gmail.com1 Brigham Young University, BYU Department of Physics and

Astronomy N283 ESC, Provo, UT 84602, USA2 Harvard-Smithsonian Center for Astrophysics, 60 Garden

Street, Cambridge, MA 02138, USA3 University of Florida, 211 Bryant Space Science Center,

Gainesville, FL 32611, USA4 Florida Institute of Technology, 150 West University Boule-

vard, Melbourne, FL 32901, USA5 Department of Physics, P.O. Box 64, 00014 University of

Helsinki, Finland6 Finnish Geospatial Research Institute, P.O. Box 15, 02430

Masala, Finland

Minor planets that originate in other planetary sys-tems but are currently unbound are usually called inter-stellar comets (ICs).7 Although there is one candidateIC known (§2.2), at present, their existence is essentiallytheoretical (Whipple 1975; Sekanina 1976; McGlynn &Chapman 1989; Francis 2005). Technically, many longperiod comets have slightly hyperbolic orbits, but theseclearly originate in the solar system and only appear un-bound at present due to minor gravitational and non-gravitational perturbations, so are not considered ICs.Indeed, identification of an object as a bona fide IC inthe usual orbit determination process would be straight-forward, since ICs would have a highly hyperbolic or-bit, i.e., eccentricities clearly greater than 1. Future ad-vanced sky surveys, particularly the Large Synoptic Sur-vey Telescope (LSST), will be many times more sensitivethan past or present observations (LSST Science Collab-orations et al. 2009). It is therefore natural to considerwhether LSST will detect ICs.

The motivation for finding ICs is two-fold: discover-ing an IC would provide new observational opportunitiesand an IC would be an in situ sample of another solarsystem. Like the discovery of the population of asteroids

7 Although there are no codified definitions, objects unboundfrom any star that are the natural minor body extension of the in-terstellar medium are usually called interstellar comets (e.g., Whip-ple 1975; Sekanina 1976), while minor bodies detected orbitingaround other stars are a natural extension of extra-solar planetsand are often referred to as extra-solar planetesimals(e.g., Jura2005). Moro-Martın et al. (2009) is an exception to this typicalnomenclature.

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∼200 years ago and the discovery of the Kuiper belt 20years ago (Jewitt & Luu 1993), the eventual discoveryof ICs will open new and unique avenues for explorationthat will improve our understanding of the formation andevolution of planetary systems. Photometric, astromet-ric, and spectroscopic investigations can reveal estimatesof the origin, physical, and chemical properties of a pieceof another solar system. Even without detailed follow-up observations, the currently unknown frequency of ICsis a useful insight into the efficiency of planet formationin the galaxy. By estimating the frequency at which weexpect to observe ICs and then comparing the expectedvalue to the actual observational frequency, we can adjustplanet formation models accordingly (e.g., Stern 1990).

No matter the particulars, discovering ICs or placingupper limits on their frequency would help us to place oursolar system in galactic context. However, as a rarely ob-served population with unique orbital properties, search-ing for ICs in LSST data will likely require a significantdedicated effort. The value of this effort depends partlyon whether the frequency of ICs detected by LSST willhave the power to discriminate between different planetformation models. For this reason, we provide a carefulassessment of the sensitivity of LSST to different param-eters of the IC population.

The assessments of the frequency of detectable ICs hasbeen highly variable (§2.1). Earlier studies predictedvery high numbers of observable ICs, usually by takingwhat was known from the formation of our solar system,estimating how many comets our solar system ejectedinto interstellar space, and then multiplying by the num-ber density of stars. For example, McGlynn & Chap-man (1989) predicted that the number density of 1 kmICs was approximately 1013 pc−3 and that this impliedthat several ICs should have already been detected. Re-cently, a careful assessment of the frequency of ICs byMoro-Martın et al. (2009, hereafter M09) showed thatthe actual mass density is orders of magnitude less thanthese previous estimates, resulting in a number densityfor ICs larger than 1 km of 105−10 pc−3. M09 is thefirst study to self-consistently account for several realis-tic properties of the IC population by incorporating thestellar mass function, giant planet frequency estimates,solar system minor planet size distributions, and a morerecent understanding of the formation of planetary sys-tems.

M09 considered the detectability of ICs by LSST, pro-viding a clear explanation of why we have not observedany ICs to date. However, their analytical investigationwas limited to considering ICs at the distance of Jupiterand beyond. They concluded that LSST would not beable to detect ICs at this distance. However, there areseveral aspects that may significantly enhance the fre-quency of detectable ICs closer than Jupiter: gravita-tional focusing by the Sun would enhance the concen-tration of ICs within Jupiter’s orbit; ICs may brightenby several magnitudes due to outgassing; much more fre-quent smaller bodies can be seen at closer distances in amagnitude limited survey; etc.

In order to estimate the realistic detectability of inter-stellar comets, we have developed a numerical simulationin order to consider all of the factors that play a role indetecting ICs. Our model includes effects from:

• increased density due to gravitational focusing (theeffect of the Sun altering the trajectory of ICs);

• photometric phase functions (the effect of observ-ing ICs at different angles );

• comet brightening (accounting for the increase inbrightness of comets as they approach the Sun);

• conditions required for observability such as solarelongation (the angular distance between the ICand the Sun) and air mass (e.g., constraints on thealtitude of topocentric observations).

These realistic factors will be discussed in full detail in§3. In addition, our method allows for a determinationof many other IC properties relevant to observers, suchas typical orbital parameters, rates of motion, and skydistribution.

It is worth noting that the results of this paper canbe divided into two parts. The orbit propagation andastrometry is based on some small assumptions, but ismostly robust. The estimation of the number of ICs thatcould be detected by LSST, on the other hand, requiresseveral assumptions, in some cases using quantities notknown even to within an order of magnitude which weleave as tunable parameters. In this regard, we occasion-ally neglect effects that would change the highly uncer-tain results by a factor of ∼2. The large uncertainty inour estimates should not be seen as a drawback of themodel, but rather a motivation to search for ICs in orderto place constraints on their currently unknown proper-ties, with implications for planet formation theory.

2. BACKGROUND

2.1. History and Results of Other IC Studies

The currently accepted model of the origin of long-period comets was not always widely accepted. Aftersome initial work by Opik (1932), the beginnings of themodern model of an isotropic cloud of comets tenuouslybound to the solar system emplaced by the planet for-mation process was originally posed as a “hypothesis”by Oort (1950). At that time, only ∼20 long-periodcomets had well-determined orbits after correcting forplanetary perturbations. One major alternative hypoth-esis was that all (long-period) comets were interstellar,with perhaps an unseen stellar companion to the Sunhelping to capture them (e.g., Valtonen & Innanen 1982).The distribution of long-period comets and other smallbodies in the solar system now give overwhelming evi-dence for cometary origins in the Oort cloud, though theorigin and history of this cloud is still under discussion(e.g., Levison et al. 2010a; Kaib et al. 2011).

One of the earliest references to ICs in the contextof the Oort cloud origin for long-period comets, is theestimate of Whipple (1975) on the frequency of ICs fromtheir non-detection and from the frequency of gamma-ray bursts. As required for an unobserved population,Whipple (1975) made various assumptions to estimatethe mass density of ICs in the galaxy to be less than3× 10−4M pc−3. Similarly, Sekanina (1976) estimatedan upper limit to the mass density of comets of .6 ×10−4M pc−3 based on the non-detection of ICs up tothat point.

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Based on updated estimates of the number of Oortcloud comets, McGlynn & Chapman (1989) use a sim-ple model to estimate that the number of comets is∼1013 pc−3. If all of these are assumed to have ∼1 kmin radius, this suggests a mass density of approximately10−5M pc−3. These estimates effectively took the num-ber of Oort cloud comets expected from the formation ofthe solar system (∼1014 at the time) and multiplied thisby the local density of stars. This is well explained byStern (1990), who explicitly consider how the frequencyof ICs can be used to infer properties of planet formationin the galaxy.

Upon finding that we have not seen the predicted num-ber of ICs, McGlynn & Chapman (1989) attempt to drawstrong conclusions, despite the simplicity of their modelfor the frequency and detectability of ICs. As a response,Sen & Rama (1993) suggested that the stellar densityused by McGlynn & Chapman (1989) was an order ofmagnitude too high, bringing the expected frequency ofdetectable ICs down enough that they were no longermissing from the observations.

Using a much more detailed model and a survey sim-ulator, Francis (2005) placed a limit of 3× 1012 ICs percubic parsec based on a non-detection of ICs in the LIN-EAR survey. If these are all assumed to be ∼1 km inradius, this suggests a mass density of 3×10−6M pc−3,two orders of magnitude lower than the earliest esti-mates. Meinke et al. (2004) mention a study of theSpaceguard survey which found a 97% upper limit on1 km ICs of 1014 ICs per cubic parsec, but included apower law distribution (corresponding to q1 = q2 = 3.5in our nomenclature below).

The most recent study (Engelhardt et al. 2014) usedPan-STARRS 1 data, a careful analysis of detectability,a power-law distribution (with q1 = q2 = 3.5), and in-cluded the possibility of cometary activity. This mostsophisticated method has similarities with our method-ology described below. The 90% confidence upper limiton the number density of &1 km ICs per cubic parsecwas 4.7 × 1014 for inactive comets (“asteroids” in ournomenclature) and 1.6 × 1013 for active comets. This isnot as strong a limit as the Francis (2005) result, but isthe result of a more detailed assessment. It is also ableto roughly rule out the McGlynn & Chapman (1989) es-timate.

M09 was the first theoretical estimate to use a moremodern understanding of planet formation, accurate dis-tributions for the frequency of stars of different massesin the Galaxy, and incorporating an IC size distributionto determine a much more realistic estimate for the fre-quency of ICs. Although their model makes many as-sumptions about the unknown properties of planet for-mation and IC properties, these are mostly included inthe form of tunable parameters. M09 estimate the massdensity of ICs to be ∼2 × 10−7M pc−3, which, whenconsidering a size distribution for ICs, yields a numberdensity of ICs larger than 1 km of 105−10 pc−3. This isbetween 3 and 8 orders of magnitude below the McG-lynn & Chapman (1989) estimate and the Engelhardtet al. (2014) upper observational limit.

Following Alcock et al. (1986), Jura (2011) has alsorecently shown that the space density of ICs must be lessthan predicted by the earlier optimistic assessments by

studying the chemical composition of unpolluted whitedwarfs (which would have retained signatures of accretedICs). They place an upper bound on the space density ofICs in agreement with M09 and far less than McGlynn &Chapman (1989). Zubovas et al. (2012) predict a similarfrequency of ICs near the galactic center as a possiblesource of Sgr A* flares.

Besides illustrating the decline in IC frequency esti-mated over time, these studies show that simplifyingassumptions and unknown properties can yield IC de-tectability estimates that span many orders of magni-tude. Even intercomparing the results of these studiesrequires assumptions about the size distribution or typi-cal size of detectable comet nuclei and the mass-radius-brightness relation of comet nuclei, neither of which areknown very well. As in M09, we have tried to use tun-able parameters to understand the importance of variousassumptions. By using the most up-to-date theory fromM09 with a model that incorporates realistic estimatesof detectability, we have produced the most sophisticatedestimate of the frequency of detectable of ICs to date.

2.2. A Candidate Interstellar Comet?

Krolikowska & Dybczynski (2013) and Dybczynski &Krolikowska (2015) find that comet C/2007 W1 (Boat-tini) is a strong candidate interstellar comet. Whilesome comets appear slightly hyperbolic due to inter-actions with the giant planets and/or non-gravitationalforces, an analysis of these effects show that, in this case,they are much too small to explain C/2007 W1’s orbitwith an initial semi-major axis of -23000 AU (1/a =−42.75±2.34×10−6 AU), perihelion of 0.83 AU, and in-clination of 10 (Dybczynski & Krolikowska 2015). Over1000 observations of C/2007 W1 were obtained over 13months, sufficient to produce a high quality orbit, as re-flected on the Minor Planet Center, which has similarorbital estimates. Another candidate interstellar comet,C/1853 E1 (Secchi), relies on century-old astrometry andcan be probably attributed to systematic errors (Bran-ham 2012).

C/2007 W1 was observed spectroscopically and foundto be an unusual comet chemically, as might be expectedfor an IC (Villanueva et al. 2011), although it wasn’tcompletely out of the range of known comets. It wasalso the source of a somewhat unusual meteor shower(Wiegert et al. 2011).

Converting the excess energy in C/2007 W1 from Dy-bczynski & Krolikowska (2015) to a velocity at infinitygives v∞ ' 0.27 ± 0.01 km s−1, much smaller than theexpected ∼20 km s−1 for an IC. Along the same lines,C/2007 W1 has a post-perihelion orbit that is bound tothe solar system (semi-major axis of 1800 AU, perihe-lion distance of 0.85 AU) which seems unusual for an IC,though this isn’t an entirely separate argument, sincethe capture is highly enhanced due to the low v∞. Inour simulations, we do not find a particularly enhancedpreference for detecting low v∞ ICs, so this low relativevelocity is seemingly an argument against the interstellarorigin of C/2007 W1.

Altogether, the orbital and chemical evidence for theinterstellar nature of C/2007 W1 is highly suggestive, butnot conclusive. We, therefore, proceed without includingthis comet in our analysis.

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2.3. Properties of the Source Region of InterstellarComets

During the lifetime of LSST (10 years), even ICs withunusually high (100 km/s) relative velocities to the Sunwill only traverse approximately 200 AU or ∼0.001 par-secs. Therefore, the dominant medium of ICs observ-able in the next several decades is the region of spaceextremely close to the Sun, in a galactic sense, and dom-inated by the so-called Circumheliospheric InterstellarMedium (CHISM or CISM), sometimes called the VeryLocal Interstellar Medium (VLISM). This region is muchsmaller than the Local Interstellar Cloud (∼10 pc) or Lo-cal Bubble (∼100 pc), and not even much larger than theheliosphere (∼100 AU). Indeed, any IC passing by Earthwithin the next decades is currently residing within theinner Oort cloud, far closer than the aphelion of Sednaand other Kuiper belt objects (Brown et al. 2004). Herewe briefly review the known properties of the IC sourceregion. In situ samples of the CHISM as it flows intothe inner solar system are taken by observing interstellarneutral atoms, interstellar pickup ions, interstellar dustgrains (.1µm, entrained in the local gas flow), and inter-stellar micrometeorites (&10µm and decoupled from thegas). Observations of the CHISM over the last 4 decadeshave not shown any significant evolution, suggesting thatthe properties are homogeneous on the scales relevant tonear-future IC detections (Frisch et al. 2009). The massratio of gas to dust in the CHISM is thought to be ∼100(Frisch et al. 1999) and the dust mass density is about∼100 times bigger than the mass in ICs estimated byM09 (Landgraf et al. 2000).

Dust smaller than ∼10 microns is observed in situby dust detectors on NASA spacecraft (e.g., Ulysses,Cassini, Helios, and Stardust) in deep space. The obser-vations show that these grains are fully entrained in thelocal interstellar flow, as expected since they are smallenough to couple to the gas, even at the low CHISMgas densities (Frisch et al. 2011). The NASA Stardustmission returned samples from its Interstellar Dust Col-lector, which was designed to capture interstellar grainsdirectly from this local flow (Westphal et al. 2014).

On the other hand, the trajectories of interstellar dustlarger than ∼10 microns is dominated by solar gravityin the region of the Sun and can travel unperturbed forhundreds of parsecs from its source region where they aredetected as Earth-impacting micrometeorites by ground-based optical and radar observations (Baggaley 2000;Murray et al. 2004; Musci et al. 2012). Very little isknown about the frequency or properties of larger par-ticles (Socrates & Draine 2009). Observations of theseinterstellar micrometeorites have detected a broad “back-ground” source, but also a significant discrete source thatmay be associated with debris-disk and planet hostingstar β Pictoris (Baggaley 2000; Lagrange et al. 2010).Murray et al. (2004) suggest that large grains may indeedbe from discrete sources. However, there is a chance thatthese interstellar meteoroids are contaminated by solarsystem particles that have reached hyperbolic velocitiesdue to planetary encounters (Wiegert 2014). Generally,these have lower velocities than the expected incomingstream of interstellar micrometeorites, but it does presenta source of confusion. See Wiegert (2014), Hajdukovaet al. (2014) and references therein for additional discus-

sion on interstellar micrometeorites and meteorites.Direct observation of ICs with LSST or other opti-

cal surveys probes the local IC population well above1-meter sizes. M09 (and our own simulations) foundthat the most detectable objects will be the smallest ICswhich, though intrinsically fainter, typically pass muchcloser than the larger ICs which more than compensatesfor their smaller area. Large (&100 km) ICs are so rare,that they are very unlikely to be seen.

We are not considering objects in the &10000 km sizeregime, as these unbound planet-sized objects (variouslycalled free-floating planets, rogue planets, interstellarplanets, or nomads) can have intrinsic luminosity andhave been studied elsewhere (e.g., Kirkpatrick et al. 2011;Strigari et al. 2012).

3. METHODS

We estimate the number of ICs that are detectable withoptical surveys. To do so, we simulate the detectabilityof billions of ICs numerically. Starting with estimates forthe physical and orbital properties of the IC population,we calculate their brightness as a function of time basedon empirical relations developed for comets and aster-oids. Potentially detectable ICs are investigated in moredetail to determine their visibility to LSST. Throughoutwe keep track of tunable parameters, noting that the un-known frequency of ICs means that our final results canvary by at least an order of magnitude. Still, we haveattempted to be accurate in our modeling, in order tounderstand the importance of various effects.8

3.1. Physical and Orbital Properties

3.1.1. Initial Position and Velocity

Our main mode of simulation assumes that ICs havebeen well-mixed and form an isotropic population.Therefore, we simulate a large cube centered on the Sunwithin which the initial positions of ICs are randomlyand isotropically generated (but see §4.10 below). Thecube is chosen to be large enough (1000 AU) that oursimulation is insensitive to edge effects.

In keeping with our isotropic assumption, for each ICwe choose a single, initial, randomly-oriented velocity(v0). To track the importance of the velocity in IC de-tectability, we choose to do many independent runs withconstant initial velocities rather than a velocity disper-sion in a single run, but this does not affect our conclu-sions. These simulations confirmed that slower ICs weremore concentrated towards the Earth and Sun (due togravitational focusing), with about twice as many cometswith v0 of 5 km s−1 coming within ∼5 AU than cometswith v0 = 30 km s−1. Over the age of the universe, evenhigh-velocity ICs will only typically intersect their ownmass if they are smaller than ∼1 cm, suggesting thatlarger particles are completely decoupled from gas in theinterstellar medium. These ICs therefore keep their orig-inal velocity with which they were originally ejected andare stirred collisionlessly in the gravitational potential ofthe galaxy, like stars, and should have a similar velocitydispersion of tens of km s−1.

The Sun is not fixed with respect to the galactic refer-ence frame of ICs. The Sun and its vicinity are rotating

8 The code is available upon request.

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together around the center of the galaxy. To isolate thelocal relative motions relevant here, a construct calledthe Local Standard of Rest (LSR) is used, which is a co-ordinate system moving around the galaxy in a circularorbit at ∼220 km/s. The Sun’s “peculiar” motion rela-tive to the LSR is not known precisely; we use a recentproposed LSR velocity for the Sun from Schonrich et al.(2010): ((U, V,W)LSR ' (11.1, 12.2, 7.3) km s−1).To keep the Sun fixed in the center of the simulation, wesubtract this velocity from each IC, which preserves thenotion that the Sun is flying through an isotropic back-ground of ICs, though they also have their own isotropicvelocity dispersion (e.g., Whipple 1975). We neglect thetiny Coriolis force due to the rotating reference frame.

The Sun’s peculiar velocity relative to the LSR is basedon the mean motion of main sequence stars assumed tobe moving together around the galaxy; the distances tothese stars is much greater than the current source regionof ICs, but the velocity of the Sun relative to the local in-terstellar wind or flow is quite similar. Though there areslight differences (usually within the error bars), the en-trained dust, the neutral helium, and the cluster of localinterstellar clouds are all traveling at nearly the same ve-locity and uniform direction relative to the Sun (Frischet al. 2011) which is slightly faster (by 6.6 km/s) and∼40 degrees offset from the direction of the Sun’s pecu-liar motion relative to the LSR. There is some evidencethat the flow direction is even changing with time (Frisch2012). However, our simulations show that the directionand magnitude of the solar velocity does not affect thedetectability of ICs and we use the LSR-relative motionfor simplicity.

3.1.2. Orbital Motion and Gravitational Focusing

Since they are not bound to the Sun, ICs follow hy-perbolic orbits. Using the aforementioned initial helio-centric position and velocity, the entire orbital path isdetermined using the standard equations and osculatingorbital elements for hyperbolic orbits in the two-bodyproblem. We do not integrate the orbits of the ICs andtherefore do not account for perturbations by the plan-ets (Torbett 1986) or non-gravitational forces (due tooutgassing, e.g., Aksnes & Mysen 2011), since these areboth negligibly small for our statistical purposes. Bycalculating their hyperbolic orbits explicitly, our simu-lation automatically accounts for gravitational focusing:the excess of ICs that will pass near the Sun due to itsgravitational influence. This is not present in the initialplacement of the ICs, but as our simulations run forwardsand backwards in time for thousands of years, the initialisotropic placement is acceptable.

We also calculate the motion of the Earth (with anarbitrary phase). From this we can determine at any time∆earth, the distance from the IC to the Earth, and ∆sun

is the distance from the IC to the Sun. Furthermore, wecan calculate the astrometric position (Right Ascensionand Declination) of each IC as a function of time, asdiscussed below.

3.1.3. Broken Power Law Size Distribution and MassDensity

The number and size of the ICs that are initially placedin the simulation cube is determined by using a brokenpower law size distribution and an overall mass density

(mtotal). The size distribution defines the number of ICswe expect to exist for a given radius. Throughout thispaper the “size” of an IC refers to the radius of the cometnucleus. The mass density is the amount of mass weexpect exists in a given volume. The size distributionand mass density are combined by assuming that all ICshave the same density (nominally 0.5 g cm−3). As aresult we can calculate the number of ICs that wouldexist in a given volume along with their respective radii.

Our nominal simulation adopts the mass density ofM09, with the total mass of ICs of mtotal = 2.2×10−7Mpc−3 = 4.5 × 1026 g pc−3 = 5.1 × 1010 g AU−3. If themass in a cubic AU were concentrated into a single objectwith the density of water, it would only be 23 meters inradius. This is ridiculously sparse, which explains whyICs have evaded detection thus far, even though their ∼4AU/year motion relative to the Sun is constantly replen-ishing the possibility of detection.

We note that, due to the uncertain nature of planetformation and the creation and ejection of ICs, the es-timate of M09 could be substantially in error. Severaleffects could increase the mass density of ICs each by afactor of a few: an updated stellar density (Garbari et al.2012), Oort cloud stripping from galactic tides (Veras &Evans 2013), ejection of Oort clouds due to the death ofstars (Veras et al. 2011; Veras & Wyatt 2012), and manyother possible effects. Since M09, the Kepler Space Tele-scope has also discovered entirely new classes of planetarysystems, calling into question aspects of planet formationtheory used to justify the estimates of M09.

Our simulations use the size distribution prescriptionssuggested by M09. In particular, the size distributionis a broken power law, with a variable break radius (rb)and different size distribution slopes on each side of thebreak. Following M09, we define differential size distri-bution slopes q1 and q2, such that n(r) ∝ r−q1 if r < rband n(r) ∝ r−q2 if r > rb. We place practical limits onthe minimum and maximum radii in our simulations (seebelow). The number densities as a function of differentq1 and q2 values for rb = 3 km are shown in Figure 1.By detecting serendipitous KBO occultations, Schlicht-ing et al. (2012) have estimated that q1 ≈ 2.8±0.1, whileFraser & Kavelaars (2009) and Fuentes et al. (2009) sug-gest that q2 ≈ 4.5 with a break radius of rb ≈ 75 kmin the Kuiper Belt (see also limits from the lack of de-tection of KBOs by WMAP, Ichikawa & Fukugita 2011).See M09 for more discussion on the possible size distri-bution relevant for ICs; here we simulate several differentpossibilities.

Our “nominal” model uses typically assumed values forthe variables that describe the properties of ICs exceptfor the size distribution, where we use a very optimisticcase. It is important to remember throughout that theobservational and theoretical estimates of parameters inour “nominal” model are sometimes controversial andoften with significant uncertainty. This reemphasizes theimportance of leaving many variables as free parameters.

Finally, we note here that there was a small error inthe number density equation as derived by M09. Theirnumber density equation (Equation 5 in M09) is onlyvalid for radii less than the break radius because of thelimits of integration used during its derivation (A. Moro-Martın, personal communication). A piecewise equationis needed to correctly define the number density for radii

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0.1 1.0 10R [km]

106

108

1010

1012

Nr>R [pc−

3]

10-9

10-7

10-5

10-3

Nr>R [AU−

3]

Fig. 1.—: Interstellar comet (IC) cumulative numberdensity per cubic parsec and per cubic AU as a functionof different power law parameters following M09. TheM09 IC mass density of mtotal = 2.2 × 10−7M pc−3 isused. We use a broken power law; shown are the distri-butions with a break radius at 3 km. This figure alsocorrects a small error in Figure 1 of M09. The colors andline types correspond to q1 = 2.0 (light blue, lowest), 2.5(red), 3.0 (blue), and 3.5 (green, highest); q2 = 3 (solid),3.5 (dashed), 4 (dotted), 4.5 (dash-dotted), and 5 (longdashed), where q1 is the differential size distribution in-dex for objects below the break radius and q2 is the samefor objects above the break radius. Notice the wide vari-ety of number densities at the smallest sizes which trans-lates to significant uncertainty in the detectability of ICs.

less than and greater than the break radius. We havemade this correction and reproduced a plot of the numberdensity of the ICs in Figure 1, showing the correction totheir Figure 1 for their nominal mass density. Since thenumber density and observability of ICs is completelydominated by the small objects, their errors above thebreak radius have no consequence for the results reportedin M09.

3.2. Calculating IC Brightness

3.2.1. Asteroid Case

Based on the above calculations, we now have the po-sition of the IC, Sun, and the Earth at any time. Deter-mining the brightness is based initially on the standardapparent magnitude equation for asteroids(Bowell et al.1989):

V = H+ 2.5[log10(∆2

sun) + log10(∆2earth)

]−2.5 log10(γ)

(1)where H is the (asteroid) absolute magnitude, relatedto the intrinsic brightness of the asteroid independent ofthe observing geometry and γ is the correction for pho-tometric phase described below. Specifically, H is thebrightness an asteroid would have if it were 1 AU fromthe Sun and from the observer and observed with theSun-asteroid-observer angle of zero (an observing geom-

etry that is only possible if the observer is at the positionof the Sun). The absolute magnitude is intrinsic to thebody and determined by radius and albedo in the aster-oid case. We use the standard definition

H =log10 (2r) + 0.5 log10(p)− 3.1236

−0.2(2)

where r is the radius in km and p is the albedo. Follow-ing M09, our nominal assumed albedo is 0.06, but thisis actually only applied to asteroids (the connection be-tween comet brightness and nucleus radius is handled ina different way).

The photometric phase angle is the Sun-IC-Earth anglewith the IC at the vertex. To encapsulate the effects ofdiminished reflection and non-uniform surface scatteringand non-zero phase angles, we follow standard methodsof using a phase function γ to adjust the brightness ofthe ICs based on its phase angle. The phase function weuse is the standard function from Muinonen et al. (2010):

γ = (1−G)Φ1(θ) +GΦ2(θ), (3)

where G is a slope parameter, Φ1 and Φ2 are basis func-tions for the phase curve (Equation 6 from Muinonenet al. (2010)), and θ is the phase angle. The value of Gcontrols how steep the phase curve is; values close to 0indicate a steep curve and values close to 1 indicate ashallow curve. We use a steep curve with G = 0.15, thestandard value used for objects with unmeasured phasecurves. For the most part, the photometric phase an-gle correction is not significant unless observations aretaken at large angles (i.e., far from opposition), when itcan drop the brightness of an object by several magni-tudes (mostly because of the smaller “day” side). Thiscan be relevant for isotropically distributed ICs whichmay be seen in non-standard geometries.

Throughout, we refer to objects that follow this pho-tometric prescription as “asteroids” although in practicethey may be dormant or inactive comet nuclei.

3.2.2. Comet Brightening

As comets approach the Sun they can become activeand have a significant increase in intrinsic brightness.There are two aspects of comet brightness to consider:how the radius of the comet nucleus relates to its bright-ness and how the intrinsic brightness grows in time asthe comet approaches the Sun and becomes more active.In both cases, we use the standard empirical methods todetermine the brightness of ICs.

To relate the radius of comet nuclei to their bright-ness, we use the comet absolute magnitude, which wecall Hc in order to distinguish it from the asteroid abso-lute magnitude, which is defined differently. The relationto determine Hc for comets is

Hc =log10 (2r)− b2

b1, (4)

where r is again the radius of the comet nucleus in kmand b1 and b2 are empirical parameters determined fromobservational data. These terms absorb the albedo termseen in Eq. (2); the asteroid case is equivalent to b1=-0.20 and b2 = 3.1236−0.5 log10(p) = 3.73 using p = 0.06.Several different values of b1 and b2 have been estimatedas noted in Table 1, therefore, we leave these as free

7

Source b1 b2Kresak 1978 -0.20 2.10Bailey & Stagg 1988 -0.17 1.90Weissman 1996 -0.13 1.86Sosa & Fernandez 2011 -0.13 1.20asteroid with albedo 0.06 -0.20 3.73

TABLE 1: List of the empirical comet parameters thatrelate comet nucleus radius with intrinsic brightness us-ing Equation 4. The influence on comet brightness fromthese parameters is given in Figure 2.

10-1 100 101

D [km]

0

5

10

15

20

25

H10

[m

ag]

Fig. 2.—: Plot of Hc for the different empirical cometnucleus size-brightness parameters b1, which controlsthe slope, and b2, which controls the offset (Table 1and Equation 4). The lines correspond to the differentsources as follows: 6% albedo asteroid (green, solid, up-per), Kresak 1978 (blue, dotted), Bailey & Stagg 1988(red, solid, lower), Weissman 1996 (magenta, dashed),Sosa & Fernandez 2011 (black, dash dotted). These illus-trate how comets are much brighter than equivalent-sizeasteroids and the significant uncertainty involved in thecomet size-brightness relations.

parameters of our model. The nominal model uses b1 =−0.13 and b2 = 1.20, the most recent estimates fromSosa & Fernandez (2011).

After determining the absolute magnitude of thecomets based on Eq. (4), an additional term is neededto model how the intrinsic brightness of comets growsdue to increased radiation from the Sun near perihe-lion. Following the standard in the comet community,we let the brightness vary as 1/∆n

sun where n is an ad-justable parameter called the photometric index (e.g.,Sosa & Fernandez 2011). When n = 2, this reduces tothe case without comet brightening. Comets are typ-ically modeled with two different values of n, one forthe pre-perihelion approach and another for the post-perihelion orbit. We use a pre-perihelion n of 5.0 and apost-perihelion n of 3.5, standard for long period cometsthought to originate from the Oort cloud, which are thebest analog for ICs in this regard (Francis 2005). Largevalues of n correspond to steeper brightening functions.

The photometric index is based on empirical observationsof comets that are generally based on detections within.5 AU and should be determined in conjunction withHc,b1, and b2 (e.g., Sosa & Fernandez 2011). In our analysis,we allow Hc to be defined by the assumed nuclear radius(r) using Equation 4 independently of any correlationthat may exist between n, b1, and b2. This affects thedirect comparison to comet studies in the solar system,but is still a reasonable approximation in the face of othersystematic errors and uncertainties in comet parameters.We also note here that the common comet absolute mag-nitude H10 is equivalent to Hc under the assumption ofa photometric index of n = 4, e.g., a brightness thatdepends on the Sun-comet distance to the fourth power.

At distances far beyond Jupiter, the photometric indexstrongly penalizes the comet magnitude in an unrealisticway. Although it does not affect our results, we avoidthis by requiring that the magnitude of an IC be alwaysless than (e.g., brighter than) the magnitude of an in-terstellar asteroid with 6% albedo. Using this piecewisedefinition we ensure that a comet is never fainter thanan asteroid of equivalent properties. A typical exampleof the brightness of a comet and asteroid in our model isshown in Figure 3.

The full magnitude equation for comet brighteningthen becomes:

V = Hc+2.5[n

2log10(∆2

sun) + log10(∆2earth)

]−2.5 log10(γ)

(5)which includes the most relevant effects. We do notconsider intrinsic brightness variations due to rotationalmodulation or outbursts and we assume that any ex-tended brightness due to a coma is smaller than the pho-tometric aperture.

3.3. Criteria for Determining Detectability

Using the above methods, we can take a specific popu-lation of ICs and determine their brightness at any time.The vast majority of ICs in the initial simulation arecompletely undetectable due to the large region of spacemodeled (1000 AU cube). To produce results that are ro-bust from small number statistics, we use a Monte Carloapproach of simulating billions of ICs in order to ensurethat a significant number (usually &100) of detectableICs are generated. This necessitates the use of renormal-ization between simulated ICs and the actual proposedpopulation of ICs, e.g., the orbital path of each simu-lated IC can represent a variable number of ICs thatwould actually be present, given the parameters of thesimulation. We also calculate IC observations over a fewthousand years and then average the non-transient por-tions of these observations in order to determine the rateof detectable ICs, even when this rate is very small.

The first step in narrowing down the large number ofsimulated ICs into those that are potentially detectableis an optimization routine that determines the minimummagnitude of a particular IC. We begin this routine withan initial guess that maximum brightness is at perihelion.In reality, there can be multiple maxima in brightnessdue to the orbital motion of the Earth. However, wedetermined that ICs that had a minimum magnitude offainter than 28 in this first step will never be detectableby LSST.

8

10-1 100 101

Distance [AU]

10

15

20

25

30

Magnit

ude [

mag]

Fig. 3.—: Asteroid magnitude vs. comet magnitudeat opposition for an object with 1 km radius. Distancerefers to the Earth-IC distance (∆earth) and the Sun-ICdistance is ∆sun = ∆earth + 1 in this illustration. Thered dashed-dotted line represents the magnitude equa-tion for a comet with a photometric index of n = 5. Thegreen dashed line is the magnitude of an asteroid. Sincethe comet brightening parameter inappropriately penal-izes the brightness at large distances, when modeling thebrightness of an IC, we use whichever is brighter (solidblack line). Both comets and asteroids are far brighterwhen they are close to the Earth/Sun, but cometary ac-tivity can increase the brightness even further.

The second step takes these potentially detectable ICsand computes their observational parameters on a muchfiner time grid that covers the possible range of observ-able times, based on the minimum magnitude found ear-lier. Every few hours, the brightness, solar elongation,airmass, and other parameters are determined. We con-sider an IC “detectable” or “observable” if it meets thefollowing criteria during at least one timestep:

1. IC magnitude less than the limiting magnitude

2. Solar elevation less than -18 (end or beginning ofastronomical twilight)

3. IC airmass less than 2 as observed from Cerro Pa-chon (future site of LSST)

Note that the latter two effects automatically require thesolar elongation to be greater than 48. This is importantsince comet brightening is extremely and unrealisticallyenhanced when the Sun-IC distance is very small. Wedo not consider the effects of the Moon, the specific ob-serving cadence, downtime, imaging fill factor, etc. Inpractice, we find that detectable ICs are often detectableover several days or weeks, so it is not likely that LSSTwill miss a substantial fraction of these objects due tothese effects, as long as the above detectability criteriaare met. We discuss the possible effects of trailing below,but otherwise assume that the survey is 100% efficient upto the limiting magnitude for simplicity. Note that the

meaning of limiting magnitude in surveys is a 50% re-covery rate, while our usage of the term implies a 100%recovery rate.

4. RESULTS

Using our model, we calculate the results of the numberof observable ICs detectable by LSST (or other surveys)for different choices of the input parameters. This is sum-marized in one number, NLSST , the number of ICs ex-pected to be realistically detectable over the LSST’s 10-year lifespan as a function of limiting magnitude. LSSTplans to have a limiting magnitude of about V ≈ 24.5,which is the nominal value we use throughout the dis-cussion. Due to larger uncertainties in other parameters,we do not consider the effects of color or specific filterchoices.

In order to consider the effects of individual param-eters, we begin with a “nominal” model using the pa-rameters in 2 and then consider the change in NLSST

due to changes in particular parameters. The results forthese models are shown in Table 3. We then describea simple set of linear equations that can be used to ex-trapolate our results to a much wider variety of differentinput parameters than shown here. Finally, we considerthe astrometric signature of detectable ICs. Though theunknown parameters of the IC population affect NLSST

by orders of magnitude, we show specific values here todemonstrate the comparison between different models.

4.1. Nominal Model

As discussed above, our nominal model uses standard(though uncertain and sometimes controversial) valuesfor the input parameters, except for the size distribution,where we use the most optimistic case. Therefore, “nomi-nal” should not be misconstrued as “best guess” and thenominal case is designed for comparison to M09. Thespecific values chosen are given in Table 2. It uses themass density of M09 and the size distribution parame-ters (q1, q2, rb) that give the maximum number of smallobjects. We use the Sosa & Fernandez (2011) radius-brightness relation and otherwise assume typical photo-metric parameters.

Most comets seen in the inner solar system have a min-imum nucleus radius of order 1 km. A common expla-nation for the lack of small comet nuclei is that theseobjects readily disintegrate and, effectively, evaporate,before they reach the inner solar system in a coherentform (Levison et al. 2002). ICs of this size may alsobe destroyed by the same mechanism in previous closestellar passages. Therefore, we consider it unrealistic toconsider active ICs with nuclei sizes below 0.1 km, andchoose our nominal model to have a minimum IC nucleusradius of rmin = 0.1 km. Even this is very optimistic,as comet nuclei smaller than ∼1 km may be far rarerthan the size distribution would suggest. Based on thismodel, we expect LSST to be capable of observing on theof order 1 detectable IC over its 10 year lifetime (Table3, Line 1). There is significant uncertainty and signifi-cant optimism in this result, but it is plausible for LSSTto detect an interstellar comet. These conclusions arediscussed further in §5.

4.2. Difference Compared to M09

9

Parameter Description Nominal Value

mtotal mass density of ICs 4.5 × 1026 g pc−3

q1 slope of the differential IC size distribution when r < rb 3.5q2 slope of the differential IC size distribution when r > rb 5.0rb the break radius of the IC number density 3 kmrmin minimum radius of detectable ICs 0.1 kmρ bulk density of IC nuclei 0.5 g cm−3

npre pre-perihelion photometric index 5.0npost post-perihelion photometric index 3.5b1 comet absolute magnitude parameter -0.13b2 comet absolute magnitude parameter 1.2v0 velocity dispersion of ICs 30 km s−1

G phase function steepness 0.15p albedo of asteroid 0.06m limiting magnitude 24.5∆earth minimum geocentric distance allowed 0 (AU)

TABLE 2: List of input parameters for the simulation and their nominal values.

# Label Difference from nominal valuesNLSST

20.5 21.75 23 24.5 25.5 26.75

1 Nominal none 0.15 0.23 0.35 0.57 0.85 1.2

2 Nominal without gravity no gravity · · · · · · · · · 0.51 · · · · · ·3 Nominal ignoring the phase function no phase function · · · · · · · · · 0.95 · · · · · ·4 Nominal using the comet density from M09 ρ = 1.5 g cm−3, · · · · · · · · · 0.20 · · · · · ·5 Nominal excluding comets within 5 AU ∆earth ≥ 5 · · · · · · · · · 0.041 · · · · · ·

6 Moro-Martınρ = 1.5 g cm−3, npre = 2, npost = 2,

· · · · · · · · · 0.0023 · · · · · ·b1 = −0.20, b2 = 3.1236, ∆earth ≥ 5,

p = 0.06, no gravity, no phase function

7 McGlynn and Chapman mass density mtotal = 4.5 × 1030 g pc−3 · · · · · · · · · 5900 · · · · · ·8 minimum size of 0.2 km rmin = 0.2 · · · · · · · · · 0.39 · · · · · ·9 minimum size of 0.5 km rmin = 0.5 · · · · · · · · · 0.22 · · · · · ·10 minimum size of 1 km rmin = 1 · · · · · · · · · 0.14 · · · · · ·11 realistic case q1 = 2.92, rmin = 1 km · · · · · · · · · 0.0012 · · · · · ·

12 Asteroidrmin = 0.01km, npre = 2, npost = 2, · · · · · · · · · 0.94 2.9 6.8b1 = −0.20, b2 = 3.1236, p = 0.06

13 slow v0 v0 = 5 km s−1 0.15 0.23 0.35 0.64 0.86 1.2

14 Kresak comet parameters b1 = −0.20, b2 = 2.10 0.21 0.33 0.55 1.0 1.42 2.1

15 Bailey & Stagg comet parameters b1 = −0.17, b2 = 1.90 0.11 0.19 0.33 0.58 0.89 1.39

16 Weissman comet parameters b1 = −0.13, b2 = 1.86 0.0045 0.023 0.079 0.20 0.37 0.76

17 small number density q1 = 2.0, q2 = 3.0 0.00006 0.00010 0.00017 0.00029 0.00041 0.00060

18 medium number density q1 = 2.5, q2 = 3.5 0.0012 0.0019 0.0031 0.0051 0.0075 0.011

19 shallow phase function G = 1 · · · · · · · · · 0.82 · · · · · ·

TABLE 3: The number of ICs the LSST could observe in its lifetime for various different cases as a function oflimiting magnitude. NLSST is the number of ICs the LSST could observe in 10 years at the listed limiting magnitude.Each row changes a few different parameters for different situations showing how NLSST changes. Due to the MonteCarlo nature of our analysis, each value has approximately a ∼10% statistical uncertainty. The parameters and theirnominal values are defined in Table 2. Note that the uncertainty in these parameters imply that the number of ICsdetectable by LSST ranges by orders of magnitude.

Although we cannot exactly reproduce the conditionsused in the analytical estimates of M09, we can approx-imate this model by using equivalent parameters: turn-ing off the mass of the Sun (which removes gravitationalfocusing), turning off comet brightening, turning off thephase function, and only considering ICs that are at least5 AU away from Earth. We still enforce the airmassand solar elevation constraints (a reduction of a factor of∼3). This yields a detection frequency of NLSST ' 0.002(Table 3, Line 6), similar to the estimate given in M09(10−2 − 10−4).

Considering each of these factors individually (Lines 2-5), we find that the largest effect in the increase of NLSST

seen in our analysis is the inclusion of ICs that comecloser to the Earth than 5 AU. Not only does this makesICs brighter because they are closer, but it makes objectsthat are smaller detectable. Since the size distributionis steep, decreasing the minimum size of detectable ICscreates almost a 10-fold increase in the number of ex-pected detections. This can also be seen in the modest

increase gained by decreasing the nuclear bulk density;since the mass density per unit volume is fixed, this hasthe effect of increasing the number of objects at a certainsize (Line 4).

4.3. Varying Parameters

A large variety of tests show that the primary determi-nant of NLSST is the number of objects at the smallestradii, unsurprising for a magnitude-limited survey of ob-jects with steep size distributions. It also means thatour results are strongly sensitive to parameters whichaffect the number density at the smallest radii. Fora fixed number density, changing the size distributionslopes q1 and q2 strongly affects the number of objects atthe smallest sizes (Figure 1) and NLSST correlates withthese changes. Similarly, increasing the total mass den-sity of ICs to, for example, the overly optimistic estimateof McGlynn & Chapman (1989), increases the detectionfrequency significantly to NLSST ≈ 6000.

We illustrate this point by changing the minimum de-

10

tectable radius in our “nominal model” from 0.1 km to0.2, 0.5, and 1 km in Lines 8-10, which decreases NLSST

from 0.57 to 0.39, 0.22, and 0.14, respectively. The use ofdifferent minimum radii mimics that of brighter limitingmagnitudes.

A much more realistic case than the “nominal” val-ues above draws from our understanding of solar systemcomets. This uses a differential nuclear radius distri-bution of q1 = 2.92 from comets (Snodgrass et al. 2011)coupled with the fact that comets with sizes smaller than∼1 km are unusually rare (e.g., rmin = 1 km). Althoughthey are not as important, we also use q2 = 4.5 andnpre = npost = 4 (to mitigate issues with the Hc-r-n dis-tribution mentioned above). With these more realisticparameters, NLSST becomes 10−3, much lower than thenominal case because of the less favorable (but more re-alistic) size distribution and cut-off. Using rmin = 0.1would still only give NLSST ' 0.005.

Given the large uncertainties in the most importantparameters, we have determined which parameters canbe considered as minor effects. Based on our simu-lations, gravitational focusing, the specific photometricphase function, the chosen bulk nuclei density, the in-trinsic IC velocity dispersion, the direction and velocityof the Sun’s interstellar motion, and the different radius-brightness relations affect NLSST at the factor of .2 levelonly (see Table 3). Figure 4 shows the effect of the dif-ferent size-brightness relations for comets and asteroids(see Table 1), before observability criteria are enforced.

4.4. Asteroid Case

Depending on the parameters of planet formation,both rocky and icy bodies can be ejected into interstellarspace (e.g., Weissman & Levison 1997; Shannon et al.2015). If we consider the case of interstellar asteroids (ordead/inactive comets) that are not subject to disinte-gration, then we can probe to much smaller sizes wherethe objects are much more frequent (rmin = 0.01). Inthis case, we move from a specific comet brightening lawto using a particular albedo, nominally 0.06. However,we lose the advantage given by comet brightening. Con-sidering the interstellar asteroid case down to a size of10 meters shows that these two effects roughly cancelwith NLSST ≈ 0.9. (Table 1, Line 12). Figure 4 showshow the comet radius-brightness relation and the aster-oid case compare, though it is important to rememberthat these assume an overly optimistic value for the dif-ferential size distribution power law index. Note alsothat we are not considering a bimodal model of ICs, butrather assuming that the entire mass density is either inthe comet or asteroid cases.

As discussed below, interstellar asteroids of these sizesare moving very rapidly on the sky. Such objects willcreate a trail of significant length, even in LSST’s short15 second exposures. To account for this trailing, weadjusted the brightness of these objects down accordingto how long their trail would be compared to the ex-pected FWHM of LSST detections (2”), effectively ap-proximating a surface brightness of the trail as if it werea point source. This amounted to a small reduction ofNLSST down to 0.75, suggesting that trailing is just be-ginning to become important. An optimal detection al-gorithm would search for statistically significant trails,even if they fell below the point source detection limit

0.01 0.1 1.0 10R[km]

10-3

10-2

10-1

100

101

NLSST

Fig. 4.—: Comet brightening has a significant impacton the number of visible ICs. The y-axis is showing thedifferential number of detections over the 10-year LSSTbaseline (compared to the values in Table 3 which arethe cumulative number of detections). Here we can seethe effect of the different parameters b1 and b2 for cometnuclei size distribution cases compared to the asteroidcase. The nominal parameters are used, including ex-tremely optimistic radius distribution differential powerlaw slopes (q1 = 3.5). The markers correspond to the dif-ferent comet brightening cases and the asteroid case asfollows: asteroid (green, ), Kresak 1978 (blue, +), Bai-ley & Stagg 1988 (red, 2), Weissman 1996 (magenta, 4),and Sosa & Fernandez 2011 (black, ). Statistical uncer-tainties due to Monte Carlo approximation errors are oforder ∼10%, which accounts for some of the variabilityshown. These are the numbers of detectable ICs beforethe observability criteria (airmass < 2, solar elevation <-18) are applied; requiring the objects to be observabledecreases the frequency significantly, especially in the as-teroid case.

(e.g., Veres et al. 2012). In the case where rmin ≈ 0.001km and NLSST ≈ 10, such an algorithm would be es-sential, though there are unavoidable trailing losses thatpresumably mitigate the ability of detecting ∼1 meterinterstellar asteroids.

4.5. Limiting Magnitude

Changing the limiting magnitude modifies the typicaldistance at which the smallest ICs can be detected; thelarger limiting magnitude of a deeper survey increases thevolume where small ICs can be detected. In practice, theeffect on NLSST is not as strong as might be expected,due to the combination of actual IC motion with respectto the Earth and the Sun, comet brightening effects, andour requirement that ICs be greater than rmin = 0.1 km.Interestingly, shallower surveys have non-negligible sen-sitivity, suggesting that existing surveys can place inter-esting upper limits on the frequency of ICs, as seen in thework of Francis (2005), Meinke et al. (2004), and Engel-hardt et al. (2014). However, keep in mind that NLSST

assumes a 10 year survey duration that is 100% efficient

11

at finding any detectable IC brighter than the limitingmagnitude, so existing ground-based surveys should bemuch less sensitive.

Another important consideration is that our frequencyof detections assumes that the IC must be detected in asingle image or exposure. It is possible that “shift-and-stack” techniques (e.g. Parker & Kavelaars 2010; Heinzeet al. 2015) can use the same survey to effectively reach∼2 magnitudes deeper, which would certainly improvethe detection rate as seen in Table 1.

The change in NLSST as a function of magnitude sug-gests that it is better to spend survey time to go “wide”than it is to go deeper, all else being equal. This ispartly due to the fact that the IC population is con-tinuously replenishing due to solar motion through theGalaxy. However, the rarity of ICs means that even a sur-vey wide enough to cover the whole sky every three dayswill still need to go deep in order to have a clear chanceof detecting ICs. It also suggests that LSST’s use of“Deep Drilling Fields” which are more heavily observedto reach a deeper co-added magnitude, are not likely tohelp unless combined with a shift-and-stack technique.

The asteroid case is much more sensitive as a func-tion of magnitude. Very deep searches that are sensi-tive below rmin = 0.01 km could significantly benefitfrom going deeper, but would have to deal with largeramounts of trailing. A new technique which combineshigh-speed cameras with the shift-and-stack techniquehas successfully discovered Near Earth Objects (whichwe show below would have similar rates of motion as ICswould have) as small as ∼8 meters (Zhai et al. 2014; Shaoet al. 2014). However, as ICs are incredibly more sparsethan NEOs, this method would have to be scaled up byorders of magnitude before ICs would be detected.

4.6. Enabling Extrapolation to Other Values ofSimulation Parameters

Through efficient programming and data control tech-niques, we were able to complete a full simulation of bil-lions of ICs in, typically, several minutes, allowing forthe computation of many possible models (Table 3). Sothat future studies can utilize and extrapolate our re-sults, we have run several models with large variationsin one or more parameters in order to see how NLSST

changes. We have found that the following multi-linearmodel gives an accurate estimate. However, we againcaution that the true values of most relevant parametersare not well known and that the prediction of the numberof ICs detectable by LSST or other surveys ranges overorders of magnitude.

Let logN = µ(~θ) × log(r) + β(~θ) where N is the (dif-ferential, not cumulative) number of detectable ICs peryear at V=24.5 where r is the nuclear radius of the ICsin km. To go from N to NLSST requires multiplying by10 (due to the 10-year baseline) and removing objectsthat are not observable (e.g., airmass ¡ 2, solar elevation¡ -18), which is a factor of 100.5−2 depending on thespecific populations.

At the small radius end of the distribution, the rela-tionship between logN and log r is approximately lin-ear (e.g., Figure 4). In that regime, we can describethe relationship between them using a slope and y-intercept, µ, and β, which are each themselves linear

µ β

constant 2.758 -30.399q1 -0.2364 3.0801q2 -0.4544 -1.167b1 -0.28 -2.17b2 -0.365 -0.802v0 0.0029 0.0043ρ 0.38 0.054npre 0.0978 0.275npost -0.48 -1.166log10(mtotal) 0.0022 0.987

TABLE 4: Coefficients needed for extrapolation to othervalues of the simulation parameters. The frequency ofIC detections per year above a brightness of V=24.5(N) can be given by a power law for small values ofthe comet nucleus radius (r), as seen in Figure 4. We

can then define logN = µ(~θ) × log(r) + β(~θ) where µ,and β are each themselves linear functions of the pa-rameters q1, q2, b1, b2, v0, ρ, npre, npost, and log10(mtotal))(in the units given in Table 2), plus a constant. Thenumber of significant figures displayed helps to ensurean accurate estimate. The values of β, which are relatedto the normalization of the power law, are somewhat rep-resentative of the importance of this parameter, e.g., thedifferential size distribution slope for small values (q1) ismuch more important than the velocity dispersion (v0).Note that rmin is not included because the result givesN as a function of size, which can then be evaluated atthe desired small size cutoff.

functions of the parameters ~θ. We describe the valuesof µ and β as a linear combination of these parameters,~θ = (q1, q2, b1, b2, v0, ρ, npre, npost, and log10(mtotal)) (inthe units given in Table 2), plus a constant. The valuesof β are somewhat representative of the importance ofthis parameter, e.g., q1 is much more important than v0.Note that rmin is not included because the result givesN as a function of size, which can then be evaluated atthe desired small size cutoff.

The coefficients of each of the parameters in ~θ in thislinear model are given in Table 4. They are obtainedby performing a least squares fit for different runs of thesimulation. A wide range of values for each parameteris used including varying multiple parameters at once,so as to make the model applicable to values beyondwhat is listed in Table 3. Although clearly not all of thefigures listed are significant, using these values yields acorrelation between this multi-linear approximation andthe full model simulation with a high Pearson correlationcoefficient (R2 > 0.95).

To illustrate, we can very unrealistically set each ofthese values equal to 10, which would give µ = 2.758 −2.364−4.544−2.8...+0.022 = −10.571 and β = −30.399+30.801−11.67...+9.87 = −39.445 which implies that N =10−39.445r−10.571, which can then be related to NLSST asdiscussed above.

Using these results, future studies should be able togenerate their own expectations for the number of de-tectable ICs by large scale surveys covering a wide rangeof observational and theoretical parameter space.

4.7. Representative Orbits of Detectable ICs

12

We have carefully created a model of the most de-tectable ICs under realistic observing conditions. Here,we present the orbital parameters of a representativesample of the most detectable ICs and interstellar as-teroids. Table 5 shows these parameters for 20 ICs and20 interstellar asteroids drawn from the nominal case.

It also serves as an ideal test population for futureIC detection algorithms, similar in spirit to the Pan-STARRS Synthetic Solar System, which also containedan estimated interstellar object population (Grav et al.2011).

In Table 5, we show the radius for the IC from oursimulation; this is a randomly-selected list of typical de-tectable IC properties from a Monte Carlo run that sim-ulated a large population of ICs. As discussed above, themost detectable ICs typically have sizes just larger thanthe minimum detectable radius rmin (here 0.1 km), sincethese are most abundant.

Detected ICs have perihelia near the Earth’s orbit, butdetected interstellar asteroids tend to have perihelia lessthan 1 AU. The distribution of orbital angles is effectivelyisotropic. ICs with lower incoming velocities are slightlyfavored, as expected from the enhancement due to gravi-tational focusing. However, they always have significantexcess velocities, so will be clearly distinguishable fromnear-parabolic comets with v∞ / 0. This reinforces ourcaution of interpreting C/2007 W1 (Boattini) as a trueinterstellar comet (§ 2.2).

4.8. Astrometric Analysis

As input to future observational efforts, we have takenthe set of the most detectable ICs shown in Table 5 andstudied their astrometry for the purpose of orbit inver-sion. We estimate the amount of observational data re-quired to determine that an object is an IC and to secureits orbit so that follow-up observations can be obtained.

To set the stage, Figure 5 shows the rate of motion ofICs the sky as a function of apparent brightness. ICshave rapid apparent motion, with typical rates of hun-dreds of arcseconds per hour, comparable to near-Earthobjects (NEOs).

First, we generate LSST-like synthetic astrometry forthese ICs. Then the resulting astrometry is fed into a sta-tistical orbit computation algorithm which provides anorbital solution. Finally we compute ephemerides basedon the orbital solution. The software tools (less the shellscripts) are available in the OpenOrb package (Granviket al. 2009) 9. This code is different from the one devel-oped to identify detectable ICs. Due to the difference inEarth’s orbital position and a few other small details, notall of the ICs in Table 5 are detected (at our magnitudecutoff of V=24.5) in these astrometric simulations, butthis does not affect the results. This completely separatecode also finds most of the ICs in Table 5 as detectable,partially validating our simulations.

For determining the astrometric orbits, we use the two-body approximation, no non-gravitational effects, JPL’sDE405 planetary ephemerides, and a geocentric observer(as opposed to the topocentric observer used above).These approximations do not affect the astrometric re-sults.

9 https://github.com/oorb/oorb

10-1 100 101 102 103 104

Rate of motion (arcsec/hr)

16

18

20

22

24

Magnit

ude (

mag)

Fig. 5.—: Distribution of sky rate of motion vs. magni-tude for detectable ICs. In this case, the ICs were givenzero velocity dispersion. The nominal and other modelsare similar. For individual objects (except for ICs whichare observed near their apparent fixed points), an in-creased brightness and rate are correlated due to a closerapproach to Earth.

Synthetic astrometry is generated by propagating theICs through their perihelion passage and recording their(RA,Dec) coordinates twice with an interval of about 15minutes every three days if the apparent V -magnitudeV < 24.5, the solar elongation ε > 45 deg, and the lu-nar elongation εMoon > 45 deg. Random Gaussian noisewith σ = 0.1′′ is added to the coordinates to mimic astro-metric uncertainty, though this may be an overestimatefor a fully-calibrated LSST.

We compute an orbital solution using the statisti-cal ranging method (Virtanen et al. 2001; Granvik &Muinonen 2005) for each night that an object is detectedtwice. The process thus resembles the operation of areal survey where the orbital uncertainty of a given ob-ject gradually diminishes as more astrometry is obtained.The ranging method provides the full non-linear orbital-element probability-density function (PDF) based on thesynthetic astrometry—in practice a cloud of weighted or-bital solutions that reproduce the synthetic astrometrywithin the limits set by the astrometric uncertainty. Us-ing the PDF we assess whether elliptical solutions can beruled out based on the synthetic astrometry.

The most obvious characteristic that separates an ICfrom its solar-system counterparts is its hyperbolic orbitwith respect to the Sun. The typically high inclinationsof ICs (Table 5) could also be utilized but this wouldpossibly lead to a confusion as high inclinations and evenretrograde orbits are known to exist for both near-Earthcomets and near-Earth asteroids originating in the solarsystem (Greenstreet et al. 2012). To unambiguously de-termine whether an object is an IC we use the criterionthat the minimum eccentricity within the orbital-elementcredibility region has to be greater than unity. In whatfollows, the credibility region encompasses 99.73% of anorbital solution’s total probability mass thus correspond-

13

Radius (km) a (AU) e q (AU) i () Ω () ω () v∞ (km/s)

0.11 -0.72625 2.4336 1.0412 60.728 109.24 331.18 34.860.18 -6.1602 1.1571 0.96798 165.16 17.084 279.55 11.970.21 -0.69227 5.5729 3.1657 121.13 307.16 167.01 35.700.11 -4.8327 1.2481 1.1988 35.800 132.42 63.724 13.510.12 -0.99266 2.2125 1.2036 83.129 355.76 118.45 29.810.11 -0.43832 4.1940 1.4000 98.972 265.91 142.86 44.870.12 -1.7647 1.7781 1.3747 55.182 8.6212 331.63 22.360.20 -0.51516 4.8583 1.9876 49.244 293.03 125.17 41.390.12 -0.51014 5.1957 2.1404 24.874 267.36 140.49 41.590.11 -0.41419 3.8027 1.1608 51.671 293.28 115.89 46.160.10 -1.4211 1.1802 0.25631 35.660 227.27 120.63 24.910.14 -0.57002 3.4843 1.4161 50.452 326.48 103.23 39.340.11 -0.42728 4.7306 1.5940 136.89 310.14 111.82 45.440.39 -0.38712 9.1568 3.1577 124.25 309.47 125.98 47.740.43 -1.1024 3.5955 2.8612 76.358 221.06 56.088 28.290.12 -0.99311 1.5962 0.59213 106.96 217.56 106.32 29.810.11 -1.9871 1.1967 0.39085 20.689 204.23 15.098 21.070.15 -0.46875 3.5816 1.2101 67.981 80.129 278.54 43.390.11 -1.5640 1.0915 0.14310 105.47 78.959 217.24 23.750.11 -0.39364 6.9736 2.3515 22.201 207.98 163.29 47.34

0.032 -0.68023 1.9241 0.62862 124.65 156.97 258.53 36.020.069 -1.6503 1.6707 1.1068 75.355 7.4526 75.896 23.120.015 -3.5974 1.0529 0.19034 68.521 54.221 124.26 15.660.017 -0.95701 1.0238 0.022733 6.1824 359.55 229.32 30.360.041 -0.53558 2.2423 0.66536 167.17 258.52 77.446 40.590.053 -0.70879 2.1711 0.83010 108.78 315.23 149.74 35.280.11 -4.1081 1.4083 1.6775 19.809 170.96 268.51 14.660.034 -0.42086 2.6194 0.68155 114.89 304.18 117.75 45.790.012 -0.90621 1.5180 0.46943 86.207 249.46 94.273 31.200.033 -1.0996 1.4696 0.51641 109.64 209.49 81.180 28.330.018 -0.45199 2.3399 0.60561 94.492 296.68 110.37 44.180.050 -1.0242 1.0939 0.096220 166.45 292.78 97.257 29.350.016 -1.5610 1.4506 0.70345 169.23 16.931 136.11 23.770.011 -0.74861 1.3012 0.22552 121.60 118.92 250.08 34.330.028 -2.7367 1.0649 0.17765 90.813 27.294 133.56 17.960.016 -5.6296 1.1988 1.1192 120.03 342.69 2.0826 12.520.017 -6.2956 1.0560 0.35259 137.30 37.675 107.71 11.840.11 -0.50025 3.9496 1.4755 167.67 243.17 63.798 41.910.018 -4.3754 1.0949 0.41516 83.139 291.61 273.74 14.200.29 -1.0272 2.6104 1.6542 39.742 280.81 157.18 29.31

TABLE 5: Orbital properties of typical detectable interstellar comets (first 20 entries) and interstellar asteroids (last 20 entries).

Standard hyperbolic orbital elements in the ecliptic heliocentric reference frame are used: a is semi-major axis, e is eccentricity, i is

inclination relative to the ecliptic, Ω is the longitude of the ascending node, ω is the argument of periapse, and v∞ is the excess velocity

at infinity.

ing to the 3σ limit of a one-dimensional Gaussian distri-bution. In astrometric analyses of solar system bodies,we have found that single night arcs can always be fitwith a hyperbolic orbit model, therefore, we require atleast 2 pairs of detections spaced by 3 nights (in our mockLSST-like cadence) before considering a hyperbolic de-tection secure. It is interesting to note that while allsolar system objects appear to be moving retrograde atopposition due to the Earth’s faster orbital velocity, ICsare an exception to this rule. Even at opposition, theycan have a prograde motion, as a result of their excessvelocity.

4.9. Comet Model

Each IC is typically detected tens or even hundreds oftimes during its perihelion passage. They are typicallyfirst “discovered” in the inner solar system at a helio-centric distance of ∼6 AU. The minimum and mediandistance to a detectable IC are ∼1.5 and ∼3 AU, respec-tively. If there were no lower limit on the size of an IC,i.e., if we allowed rmin . 0.1 km, then we would detect

a larger population of smaller bodies closer to the Earth(see the astrometric results for the asteroid case below).

Most ICs require 2–3 independent nights of observa-tions during a timespan of 4–7 days to ensure that ahyperbolic orbit is the only viable solution. This rapididentification is due to the high rates of motion seen fortypical ICs (∼200 arcseconds/hr or ∼1 deg/day), allow-ing for precise astrometric constraints over a short ob-servational arc. The rapid motion is partially due to thehigher orbital velocity experienced by unbound objects,but is also strongly due to the fact that a magnitude-limited survey will identify nearby objects that will haverapid apparent motion due to parallax from the Earth’smotion. When the data are insufficient for a strong clas-sification, ICs are usually confused with Aethra asteroids(Mars crossers).

In order to facilitate follow-up observations we com-pute the maximum sky-plane uncertainty 3 days and 14days after the last detection. We assume that, e.g., pho-tometric, spectrometric and polarimetric follow-up ob-servations are feasible when the ephemeris uncertainty

14

is below 30” 3 days after the last observation. Whenthis limit has been reached it essentially guarantees thatthe uncertainty does not grow with time as the ongoingsurvey provides additional astrometry every 3 days. Sim-ilarly, we assume that Target-of-Opportunity-type obser-vations can be planned when the ephemeris uncertaintyis less than one degree within 2 weeks after the last de-tection. Most objects require 3 nights of observationsspanning 7 days for the above criteria on the ephemerisuncertainty to be fulfilled. That is, the timeframe for es-tablishing that an object is an IC is the same as the timeframe for establishing its orbit sufficient for follow-up.

The ephemeris uncertainty drops quickly with increas-ing astrometry: at the time of fulfilling the above cri-teria the typical 3-day ephemeris uncertainty is only afew arcseconds, a few tens of arcseconds for the 14-dayephemeris uncertainty, and less than 0.2 30 days intothe future.

4.9.1. Asteroid Model

The astrometric results for ICs are probably applicablefor a wide range of possible populations. However, inter-stellar asteroids are only detected in abundance whenmuch smaller objects are seen and these ∼10-meter ob-jects must necessarily pass very close to the Earth to bedetected by LSST. As this can strongly affect the astro-metric solutions, we repeated the above analysis for theasteroid population given in Table 5.

Unsurprisingly, the typical geocentric discovery dis-tance was nearly always less than 1 AU and the num-ber of detections per perihelion passage was much lower,usually less than 10. These smaller bodies require morefavorable observational circumstances to be brighter thanthe limiting magnitude.

Otherwise, the asteroid case was similar to the ICcase: 2–3 nights spanning 4–7 days was sufficient toidentify objects as hyperbolic and to have an ephemerisuncertainty small enough for recovery and Target-of-Opportunity observations. When the tracklet was tooshort to securely identify the object as interstellar, themost common classification was as an Apollo NEO.

4.10. Possible Discrete Source of ICs

Throughout the analysis thus far, we have been con-sidering an IC population that is completely isotropic.However, it is possible that the population of ICs ismore heterogeneous. If we were passing through the Oortcloud of another star, for example, the IC density wouldgo up significantly (Stern 1987). Recent stellar passagesmay be effective at temporarily stripping the Oort cloudsof other stars, eventually resulting in an anisotropic ICsource (e.g., Zheng & Valtonen 1999).

ICs from individual systems are ejected primarily atinclinations less than ∼30 with respect to the invariableplane (e.g., Duncan et al. 1989). However, these planesare oriented isotropically, so that the IC “luminosity” ofa planet forming system is similar (proportional to theinverse square of distance) to conventional photon lumi-nosity. Throughout, we’ve assumed that we are search-ing for ICs from, effectively, the “diffuse IC background”.Here we consider the possibility of a single dominant dis-crete source of ICs.

As pointed out in §2.3, interstellar micrometeorites ap-pear to have a discrete source possibly associated with

edge-on debris-disk and planet hosting star β Pictoris(Baggaley 2000; Murray et al. 2004). A similar sourcefor ICs is plausible and may significantly enhance theirfrequency over the estimates of M09.

As one of the nearest forming stars (19.4 pc), β Pic hasbeen extensively studied. Since the discovery of microm-eteories, observers have detected a collisionally activemulti-component debris disk (Golimowski et al. 2006; deVries et al. 2012) and a directly imaged planet (Lagrangeet al. 2010; Chauvin et al. 2012) that was earlier pre-dicted by theorists (e.g., Freistetter et al. 2007). Thereis every reason to expect that β Pic is actively ejectingICs, some of which are currently passing through our so-lar system; the larger brethren of the already detectedinterstellar micrometeorites. (It’s systemic radial veloc-ity is ∼20 km s−1 and therefore cannot be a source ofC/2007 W1, which was verified by direct backwards in-tegration (P. A. Dybczyski, pers. comm.).)

To investigate this possibility, we considered a modelwhere the ICs had no intrinsic velocity dispersion andthus come streaming in due only to the Sun’s relativemotion. This simulates what a single population of ICsfrom a discrete source may look like. Gravitational fo-cusing can cause enhancements of interstellar particlesat the antapex of the Sun’s motion, and we did detecta weak spatial clustering of IC detections in this case.(Our nominal model showed no spatial clustering, as ex-pected when the velocity dispersion of the ICs is largeror comparable to the solar velocity.)

The lower relative velocity also increases the impor-tance of gravitational focusing and results in a slightlylarger number of detectable objects to NLSST of 0.61, allelse being equal. A “high IC luminosity” discrete sourcemay also enhance the mass density of ICs, which is notincluded here, but could easily be a significant effect.

It is tantalizing to note that if an IC is detected andits orbit recovered, backwards integration over severalmillion years could reveal a very specific location for itsoriginal source, potentially identifying it as a planetes-imal from a specific system like β Pic (Dybczynski &Krolikowska 2015). Note that β Pic is here used as anexample; the actual IC population may come from otherdiscrete sources, including, potentially, multiple sources.

5. DISCUSSION AND CONCLUSIONS

The likelihood of detecting interstellar planetesimalshas had a long and varied history. A modern understand-ing of the properties of the IC population was recentlyproposed by M09. In this work, we’ve studied the realis-tic observational aspects of detecting and characterizingthis unknown population. Using a numerical model thattracks the position and brightness of ICs, we estimatethat LSST could detect on the order of 1 IC during its10 year lifetime, with orders of magnitude uncertaintymostly based on the actual frequency of small ICs. Theexpected size distribution of objects reduces this rate to∼0.001, but including the contribution from interstel-lar asteroids or comet outbursts or discrete sources mayboost the detection by 1-2 orders of magnitude. Frankly,some optimism is required to conclude that LSST willdetect even 1 interstellar object.

While it is possible to improve our model, our resultsare sufficiently informative to begin the discussion ofwhether and how the astronomical community should

15

conduct the search for ICs. Facing the stark realiza-tion that ICs are exquisitely rare, we can expect to findthem at the threshold of detection. In a single night,they are generally indistinguishable from NEOs, aster-oids, and long-period comets. Only at the time of thenext LSST observations (nominally 3 and 6 days later),will it become clear that the orbit can only be fit whenthe eccentricity is greater than 1. They are movingrapidly (∼200 arcsec/hr, ∼1 deg/day) and will be dif-ficult to link between single night detections. It is alsolikely that there will be occasional false positive linkagesbetween unrelated solar system bodies that initially ap-pear to be ICs. Algorithms attempting to detect solarsystem bodies may choose to discard detections and/orlinkages that indicate an unbound orbit. This may be aneasy way to help make the solar system moving objectsearch more tractable, even though it would throw awayany ICs that could nominally be detected. If possible, werecommend that systems searching for NEOs, asteroids,and/or comets (such as the Pan-STARRS Moving Ob-ject Processing System; Denneau et al. 2012) refrain fromexplicitly or implicitly biasing their systems against thealgorithmic detectability of ICs, though these are vastlyless frequent than solar system small bodies. The Pan-STARRS MOPS, in particular, is not explicitly biasedagainst hyperbolic orbits.

ICs convey rare and unique planet formation informa-tion; rare because ICs are so hard to observe and uniquebecause their observations complement other methodsused to study planet formation. The work needed todiscover ICs is accompanied by a strong desire to followthem up with other observations, both for orbit recov-ery and for detailed characterization (e.g., with JWST).

For this reason and based on our simulations, the idealcase is to discover and track ICs within 1-4 weeks, similarto NEOs. However, one of the strongest pieces of infor-mation gained from discovering an IC is their frequencyand this could be determined in a specialized post factosearch, well after any detected ICs are recoverable or ob-servable. Given the significant probability that LSST willnot detect any ICs, such a project should be prepared toplace an upper limit on the IC frequency based on a nulldetection.

While it seems difficult to imagine now, we look for-ward to the day — perhaps in the distant future — thatICs are detected in such abundance that, like KBOs andexoplanets, the number of objects rapidly grows fromzero to one to ten to a population so large it is hard tokeep track of individual objects. It is exciting to considerwhat this future regime of IC studies could reveal aboutthe formation and evolution of planetary systems in theGalaxy.

We thank Avi Loeb, Amaya Moro-Martın, Luke Dones,Mike Jura, Priscilla Frisch, and Dan Green for discus-sions and suggestions that improved the manuscript. Wethank Paul Weissman for very helpful reviews. NVCacknowledges support from the BYU Department ofPhysics and Astronomy mentoring support fund. Wethank the Division for Planetary Sciences HartmannTravel Grant Program for sponsoring a presentation ofthis work at a DPS Meeting. DR thanks Ed Sittler andthe NASA Academy for support on an earlier relatedproject. DR also acknowledges the support of a Har-vard Institute for Theory and Computation Fellowship.MG was funded by grant #137853 from the Academy ofFinland.

APPENDIX

OTHER METHODS FOR DETECTING INTERSTELLAR COMETS

There may be other methods for detecting the signatures of interstellar comets besides standard direct photometricobservation that we have assumed above. Here we briefly consider a few other possibilities, using the nominal massdensity from M09. In considering the likelihood of detecting interstellar planetesimals, it is good to remember thatthey are expected to be many orders of magnitude less common than the small body populations of the solar system.

While we have a basic model for comet brightening based on empirical observations, it is known that some cometsoccasionally undergo huge outbursts, increasing in brightness by several magnitudes (e.g., Comet Holmes). Unless ICshave some preference for these rare outbursts, the frequency of rare outbursts on rare ICs must be negligibly small.Another method for observing comets is to observe them approaching the Sun. This is very fruitful for sungrazingcomets, like those from the Kreutz group, but is not likely to be profitable for searching for rare ICs since the volumeof space that is very close to the Sun is too small.

Interstellar comets would leave meteor shower trails like regular comets. Indeed, interstellar comet candidate C/2007W1 caused a readily detectable meteor shower from its single passage through the inner solar system (Wiegert et al.2011). Passing ICs could go undetected but cause streams that may be intercepted by the Earth, but generallythese would be a tiny fraction compared to streams caused by solar system bodies. In fact, there can be hyperboliccomponents to meteors caused by gravitational interactions in the solar system, adding to the confusion (Wiegert2014). Previous interactions between ICs and other stars would have left trails that lace and thread the galaxy, butthese presumably have a short lifetime, rendering detection and characterization unlikely. There may be signs of ICaccretion onto solar system bodies. ICs can have unusually high orbital velocities and would often create hypervelocitycraters on practically any solar system surface. However, the velocities are not expected to be so high as to beotherwise inexplicable for bodies in the inner solar system. In the Kuiper belt, where typical impact velocities betweenKuiper Belt Objects are only ∼1 km/s, an impact by an IC with a velocity of ∼25 km/s could produce a somewhatunusual crater, although to first order only the impact energy can be deduced from a crater and not the initial velocity.Collisions between Kuiper Belt Objects and Oort Cloud comets can have higher collision velocities ∼5 km/s. However,even the most favorable IC population from M09 would suggest that the largest IC to hit Pluto over the age of thesolar system was ∼1 meter in radius, which would make a crater far too small to be detected by New Horizons (Weaver

16

et al. 2008). The IC accretion rate and the integrated accretion are both so much smaller than accretion from solarsystem material that even extreme chemical or isotopic differences would be washed away. For example, based on thehighest M09 abundance, the largest IC that has ever hit the Earth (which has intercepted a volume of ∼60 AU3 overits 4.5 GYr history) is ∼30 m in radius, not even Tunguska-size. The dependence on the tiny mass density of ICs andsmall cross-sections of the planets echo studies that conclude interstellar panspermia is very difficult, at best (e.g.,Melosh 2003; Wallis & Wickramasinghe 2004; Belbruno et al. 2012).

Some ICs are gravitationally captured by the solar system (Valtonen & Innanen 1982). Torbett (1986) studythe ability of Jupiter to capture ICs and conclude that these captures would only occur every ∼ 60 MYr using theunrealistically high density of McGlynn & Chapman (1989). Using estimates of M09, the total volume of planetesimalscaptured in this way over the age of the solar system is ∼1 km3. However, the analysis of Torbett (1986) assumeda single 20 km/s characteristic velocity for ICs and did not consider either a velocity dispersion, chaotic temporarycaptures, tidal disruption, or a possible discrete source of ICs. Nor did Torbett (1986) consider the post-capture orbitalevolution of ICs. Since the new heliocentric orbit intersects the orbit of Jupiter, captured ICs will not generally bedynamically long-lived (and could be re-ejected). Recent captures that have not yet been destroyed or re-ejected mayhave unique heliocentric orbits, though the largest bodies on these orbits are probably only tens of meters in diameteror smaller, making them too small to detect except in extraordinary circumstances. Intriguingly, there are knowncomets (e.g., 96P/Machholz) with unusual chemical abundances and orbital parameters that have been hypothesizedto have an interstellar origin (Langland-Shula & Smith 2007; Schleicher 2008), though the expected abundance fromM09 makes this very unlikely. Our model could fruitfully be expanded to include Jupiter and its gravitational influenceto investigate this possibility in more detail, which we leave to future work.

Things can be different in the early solar system. First, the initial proto-solar nebula was presumably seeded withthose ICs from previous generations of planet formation that had either a low velocity relative to the nebula by chanceto be captured by aerodynamic drag. These are the larger components of “pre-solar grains” found in meteorites.Comparing the M09 estimated density of ICs with the density of the ISM, suggests that roughly 10−6 of the material(by mass) in the proto-solar nebula may come from ICs. This may be sufficient to affect some processes of planetformation, which usually assume a pure dust and gas initial composition; an extreme example is that ICs may serveas seed particles to break the grain-grain bouncing growth barrier (Windmark et al. 2012). In addition, a minisculefraction of primitive small bodies in the solar system (and the ∼50000 meteorites in various collections) may actuallyhave an extra-solar provenance, though it would be very difficult to prove this conclusively. Birth in a stellar clustermay significantly enhance the probability of capturing ICs into the early solar system (e.g., Perets & Kouwenhoven2012; Levison et al. 2010b), though this would decrease any distinguishing chemical features assuming the proto-clusterto be relatively homogeneous. Indeed, it is not even clear that these would be considered “extra-solar” at all.

Another possible detection method is serendipitous occultations. At first, this seems hopeless; the Kuiper belt isfar more dense than the population of ICs, but there have only been clear detections of 2 small KBOs (Schlichtinget al. 2012), and even these are now severely called into question by New Horizons crater population statistics. Onthe other hand, presumably ICs are distributed throughout the galaxy, allowing for the volume along a line of sight toa distant object to potentially compensate for the low density of ICs. More distant objects have a larger IC “opticaldepth”, resulting in something like Olbers paradox (in the geometric optics limit, which does not generally apply,see Heyl 2010). It can be shown that the geometric enhancement in size for closer objects is exactly canceled by thesmaller number of these objects in the cone-shaped line-of-sight to a distant object and that (for q1 > 2), smallerICs dominate over large ICs in optical depth. For objects even ∼10 kpc away using the density of M09, the coveringfraction by cm-size ICs is only roughly 10−7. Extrapolating to smaller ICs becomes effectively the same as standarddust extinction towards distant objects. In any case, occultations are not a viable method for detecting the frequencyof ICs or individual objects.

It is fair to conclude that the most likely method for detecting ICs is direct optical observation of the continuouslyinflowing hyperbolic IC population as discussed in the main text above.

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